Graph Theory - Rankings in a Tournament

Question

Given a tournament graph like this : How do you find out how many rankings there are systematically? Help / explanation would really be appreciated.

Answer 1

Did you mean this graph specifically, in which every node has two "in" edges and two "out" edges? That is, a graph in which there's no reason to prefer any specific node, and any arrangement of five letters is equally good? That's just asking how many different ways you can arrange five letters.

As an aside, the graph reminds me very much of the win-lose graph of Rock, Paper, Scissors, Spock, Lizard. In fact, you get this graph if you identify x with paper, y with lizard, v with scissors, w with Spock, and u with rock. There is an arrow from a to b if a defeats b.